Here is a page from a photographic website with some Serious work on it. This
guy isn't just taking photographs, he is a Photographer, who sells his
photographs, and clearly defines his standards of use, also backs his standards
up by using tracking technology. Here is a page from the site:

For someone like him, who expects this kind of rigid adherence to copyright law
even though his stuff is on the web, his example needs to be followed, i.e., I
do not believe that this level of rigidity should be implied, well, most people
are not going to follow it naturally, but if stated outright, with possible
consequences delineated, most people will probably respect the limitations
imposed.

This message is two-fold, connecting with two of your discussion areas. The
second part of my message goes back to your September 9 entries titled
"Improving Math Education" and "Another Opinion", and gives an example to my
response about aesthetic connections with mathematics.

Imagine students walking in to their math class. The instructor has a projector
with web connection and goes to the above site, brings up a picture. It will
blow the kids minds, they'll say, what does this have to do with math. The
instructor asks the class, can you identify anything in this picture that can be
described using math? Well, unless there is some Euclidean shape in 2D or 3D
there, the students will look and say Nothing. Then you point out the clouds or
tree branches, the similarity, how the big pieces and small pieces have the same
shapes. And this without yet touching the idea of an Interated Function System.
Let the students BE, WITH the nature and internalize the similarity. Go to
another picture, and say "what can be described in this picture using math?"
Maybe it has the same thing, and they recognize the branches. The instructor
asks, what else? And maybe there are beautiful clouds in the picture. It
doesn't matter how young or old these students are. ASK THEM, what would happen
if we took a piece of that cloud and set it to the side, what it look like? And
they'll say: a cloud! That's the point, a piece of a cloud looks like a cloud.
Stay away from talking about an IFS. Let them enjoy the idea of what they are
seeing and perhaps experience a moment of a Mental Dawn with regard to their
perception of mathematics. It may happen immediately, or it may take a scroll
through many pictures and identifying similarity in nature, along with the
reinforcement of words that these natural objects can be described
mathematically, before someone asks "How?" Then the opening has been made by
the student(s) to bring up the idea of an IFS and fractal geometry.

If you have the time to look at this guy's work, it is absolutely magnificent,
and such a powerful sensory impact is really needed, that is almost enthralling.
The point of this kind of exposure and making the connections with math wouldn't
be just to expose students to concepts of the IFS and fractal geometry. The
really important thing would be to reinforce the connection so that when they
leave the classroom and see a tree outside, or look at clouds and mountains,
they will use their vision to start noticing the similarity in the nature around
them with a background realization that this is a connection with mathematics.
We're surrounded by structure in nature that can be described using mathematics
in all kinds of ways. From the time we could see the nature around us, the
structure behind it gently passes over our heads and what is perceived is the
beauty. To contribute to bringing about an awareness in students to the
structure behind that beauty is not only to educate them about math but to give
them a beautiful gift that can be enjoyed and empirically verified almost
everywhere outside, without regard to age, or social status, or financial
standing, no equipment is required except our eyes and mind.

I'm not going to go back and read this. It will probably be embarrassing. Hope
it makes sense.

This material is based upon work supported by the
National Science Foundation under Grant DUE-0226284.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s)
and do not necessarily reflect the views of the
National Science Foundation.